If you are set on working in a language school rather than a high school, choices are few and far between. As far as I know there are only two well run schools in the city. The English Academy, and Address School of English.

Yes, it is true, those two places,they are generally considered to be the best, which is not saying very much. Don't know many people who stay at these places more than a year or two, must be a good reason for that!
Ever considered working at a real school?

I know people who have worked at both schools and The English Academy has a very low staff turnover (probably due to a lack of alternatives). Address hires largely free lance teachers so I'm guessing their turnover rate is higher. There are some good private high schools in the city, too. Çakabey and Gelişim are two of them that hire native speaker teachers but there are others. There are also private universities such as Yaşar and the Economics University, the latter being a better choice.

Depending on how much experience you have you should be careful that you choose a language school with a good teacher support structure. Many of the language schools here are just happy to give you a book and then tell you to get on with it. Also try to speak to teachers who have worked at the schools. Do they really organise your work visa and resident's permit. Do they pay on time etc.

English Academy do pay on time and do organise work permits etc very efficiently (or they certainly used to when I worked there). The management are highly experienced and very supportive. This was my first job in TEfL a few years ago and I believe I couldn't have been luckier than to start at this place. I have no reason to suspect things have changed as the Manager/Owner and DOS have not changed since my day. Also was briefly at Address - lovely facilities, lovely people but as a previous poster mentioned, they tend to be more piecemeal when it comes to actual hours allocated. Good luck with the job hunt. Oh and by the way in my humble opinion Izmir is a great place to live!